Working Paper
Series
_______________________________________________________________________________________________________________________
National Centre of Competence in Research
Financial Valuation and Risk Management
Working Paper No. 231
Real Options and Risk Aversion
Julien Hugonnier
Erwan Morellec
First version: June 2003
Current version: March 2005
This research has been carried out within the NCCR FINRISK project on
“Credit Risk”
___________________________________________________________________________________________________________
Real Options and Risk Aversion∗
Julien Hugonnier
University of Lausanne
FAME
Erwan Morellec†
University of Lausanne,
University of Rochester,
CEPR and FAME
This version March 2005
First draft: June 2003
Abstract
In the standard real options approach to investment under uncertainty,
agents formulate optimal policies under the assumptions of risk neutrality or
perfect capital markets. Although the assumptions of risk neutrality or market completeness are crucial to the implications of the approach, they are not
particularly relevant to most real-world environments where agents face incomplete markets. In this paper we extend the real options approach to incorporate
risk aversion for a general class of utility functions. We show that risk aversion
increases the option value of waiting and leads to a significant erosion in project
values.
Keywords: Risk aversion; Real options; Investment timing.
JEL Classification Numbers: G11; G12; G31.
∗
We thank Mike Barclay, Cliff Smith, Jerry Zimmerman, and seminar participants at the University of Rochester for helpful comments and NCCR FINRISK of the Swiss National Science Foundation for financial support.
†
Corresponding author: Institute of Banking and Finance, Ecole des HEC, University
of Lausanne, Route de Chavannes 33, 1007 Lausanne, Switzerland.
E-mail address: [email protected].
1
Introduction
Since the seminal contributions of Brennan and Schwartz (1985), McDonald and
Siegel (1986) and Pindyck (1988), the literature analyzing investment decisions under
uncertainty has developed substantially.1 In this literature, investment opportunities
are analyzed as options written on real assets and the optimal investment policy is
derived by maximizing the value of the option to invest. Because option values depend
on the riskiness of the underlying asset, volatility is an important determinant of the
value-maximizing investment policy. As a result, one should expect attitudes toward
risk to significantly affect investment policy. Despite this observation, models of
investment decisions under uncertainty typically presume that decision makers are
risk neutral or that markets are complete and frictionless, so that decisions are made
in a preference-free environment.
Although the assumptions of risk neutrality or market completeness are crucial to
the implications of the approach, they are not particularly relevant to most real-world
environments. Indeed, while large shareholders may be able to perfectly diversify their
wealth, corporate executives and entrepreneurs typically face incomplete markets and
are exposed to undiversifiable risks. This exposure may arise for different reasons. For
example, it may arise because the cash flows from the firm’s projects are not spanned
by those of existing assets; this is typically the case of R&D projects or more generally
of investment in new lines of business. It may also arise because of compensation
packages that constrain managers to hold very undiversified positions in their firm’s
stock. Finally, it may simply arise because of transaction costs or other types of capital
market imperfections. As a result of these frictions, decision makers are typically
exposed to undiversifiable risks. Their investment decisions should therefore depend
on their attitudes towards risk. This then begs the question: How does risk aversion
affect investment decisions when undiversified corporate executives or entrepreneurs
have control rights over investment policy?
This paper develops a utility-based framework that answers this question. To
make the intuition as clear as possible, we construct a simple model of investment
decisions that builds on earlier work by McDonald and Siegel (1986). Specifically,
1
Dixit and Pindyck (1994) provide an excellent survey of this literature. See Moel and Tufano
(2002) for empirical evidence.
1
we consider an entrepreneur having an exclusive access to a project that generates a
continuous stream of cash flows. In the paper, the entrepreneur has perpetual rights
to the project and seeks to determine the investment date that maximizes his indirect
utility of wealth. Investment arises when the subjective valuation of the project equals
the full investment cost, which includes the subjective value of the option to invest.
The analysis in the paper reveals that risk aversion has a large impact on investment policy and project value. Notably, the difference in project value under firmand utility-maximizing policies can reach 20% for reasonable parameter values. As
shown in the paper, this erosion in values arises because the decision maker has a
strong incentive to delay investment, as manifested by his decision to select an investment threshold that is too high relative to the value-maximizing threshold. The
intuition underlying this result is as follows. By investing, the decision maker transforms a safe asset into a risky one. This exposes him to undiversifiable cash flow risk.
The associated increase in the volatility of consumption leads to a reduction in his
indirect utility, which in turn provides him with an incentive to delay investment.
This paper relates to a number of paper in the literature. Parrino, Poteshman
and Weisbach (2002) examine distortions in investment decisions when a new project
changes firm risk. In their setup, a risk-averse manager can invest at time zero in a
project that affects cash flow volatility. When making this investment decision, the
manager maximize his expected utility at some future date. The major difference
with this paper is that Parrino, Poteshman and Weisbach are interested in the size
of investment whereas we are interested in its timing. In particular, we consider a
dynamic investment problem in which the timing of the investment decision maximizes
the utility of the manager, whereas in their setup timing is exogenous.
Miao and Wang (2004) and Henderson (2004) examine the impact of risk aversion
on investment decisions when agents have an exponential utility function that rules
out wealth effects (instead of a general class of utility functions as we do in this paper).
Both papers allow partial hedging of cash flow risk but reach different conclusions. In
their paper, Miao and Wang find that risk aversion delays investment. However, their
solution relies on complex numerical techniques and they do not bound the support of
asset prices, therefore allowing for negative asset prices. By contrast, Henderson finds
that risk aversion speeds up investment. However, she considers a very specific setting
2
in which the benchmark risk-neutral case is degenerate. Another closely related paper
is Hugonnier, Morellec and Sundaresan (HMS, 2005) that examines the asset pricing
implications of growth options in a general equilibrium production economy. In their
model, HMS allow the representative consumer to affect the dynamics of the state
variable (wealth) by changing his consumption rate. When the growth option is very
attractive, the consumer can reduce his rate of consumption to a level that is less
than the long-run equilibrium value to accelerate the exercise. They show that the
impact of the growth option on the consumption rate of the representative consumer
depends on his degree of risk aversion, leading to a negative relation between the
option value of waiting and the coefficient of relative risk aversion.
The remainder of paper is organized as follows. Section two describes the model.
Section three analyzes investment decisions. Section four discusses implications. Section five concludes.
2
Model and assumptions
This paper analyzes the impact of risk aversion on firms’ investment decisions. For
doing so, we consider a simple generalization of McDonald and Siegel (1986) in which
an entrepreneur has an exclusive access to a project that generates a continuous
stream of cash flows after the investment date. We assume that these cash flows
are not spanned by those of existing assets. As a result, the entrepreneur faces
incomplete markets. Before the investment decision, the entrepreneur has wealth I
that is invested in a risk free technology yielding an instantaneous risk-free rate r > 0.
At any time t, the manager can invest his wealth in a risky project. As in McDonald and Siegel (1986) we consider that the investment decision is irreversible.
Moreover, we assume that the project is infinitely lived and generates an instantaneous cash flow stream X that is governed by the diffusion:
d Xt = µXt dt + σXt dZt ,
X0 = x > 0.
(1)
In this equation, µ and σ > 0 are constant and (Zt )t≥0 is a standard Brownian motion.
This equation implies that the growth rate of cash flows is Normally distributed with
mean µ∆t and variance σ 2 ∆t over time interval ∆t. (Below we place restrictions on µ
and σ to ensure that the optimization problem of the decision maker is well defined.)
3
Throughout the paper, the preferences of the decision maker are represented by
the functional
∙Z ∞
¸
−ρt
c 7−→ E
e U (ct ) dt ,
0
where U is his utility function and ρ is his time preference rate. Below we assume
that U (.) is increasing, concave, and once continuously differentiable. Thus our model
can accommodate any of the standard CARA, CRRA, and HARA utility functions.
Moreover, we consider that U is defined on a domain that includes R+ .
3
Real options and investment timing
3.1
The benchmark case
We first review the problem studied by McDonald and Siegel (1986) as presented
in Dixit and Pindyck (1994, Chapter 6). Assume that agents are risk neutral (we
could also assume that they have access to complete markets) and that the subjective
discount rate satisfies ρ = r. In that case, the objective of the decision maker is to
determine the investment policy that maximizes project value. By investing in the
project, the decision maker gives up a risk-free cash flow stream rI and gets in return
a risky cash flow stream X. Thus, his problem is to select the investment time τ that
solves the following problem:
∙Z τ
¸
Z ∞
−rs
−rs
v (x) := sup E
e rIds +
e Xs ds .
τ
τ
0
This optimization problem can also be written as
v (x) := I + F (x) .
where
F (x) = sup E
τ
∙Z
∞
−rs
e
τ
¸
(Xs − rI) ds ,
denotes the value of the investment opportunity, E denotes the expectation operator,
and τ is the unknown future time of investment.
Because X is a sufficient statistic for the investment surplus and this surplus is
increasing in X, the value-maximizing investment policy takes the form of a trigger
4
policy that can be described by a first passage time τ of the state variable (Xt )t≥0 to
a constant threshold X ∗ . Standard calculations give
¶³ ´
µ ∗
x β
X
−I
,
(2)
F (x) =
r−µ
X∗
where β > 1 is the positive root of the quadratic equation:
1 2
(3)
σ ξ (ξ − 1) + µξ − r = 0.
2
It is then immediate to show that the value-maximizing investment threshold satisfies:
X∗
β
=
I.
r−µ β−1
(4)
Equation (2) shows that the value of the investment project (F (·)) equals the
product of a stochastic discount factor ((x/X ∗ )β ) and the investment surplus (S (X ∗ )).
This discount factor accounts for both the timing of investment and the probability of
investment. Equation (4) gives the critical value X ∗ at which it is optimal to invest.
Because β > 1, we have X ∗ > (r − µ) I and it is optimal to invest when the project’s
NPV is strictly positive. Thus, irreversibility and the ability to delay lead to a range
of inaction even when the investment surplus S is positive. We now turn to the
analysis of investment decisions when the decision maker is risk averse.
3.2
Investment timing and risk aversion
While the assumptions of risk neutrality or market completeness are convenient to
characterize investment decisions under uncertainty, they are not particularly relevant
to most real-world environments. In particular, corporate executives and entrepreneurs typically have to make investment decisions in situations where the cash flows
from the project are not spanned by those of existing cash flows or under other constraints, like portfolio constraints, that make them face incomplete markets.2 In such
environments, we can expect their risk aversion to affect firms’ investment decisions.
2
These liquidity restrictions can be imposed on executives for legal reasons (SEC Rule 144). They
can also be imposed by contract (lockup periods in IPOs or M&As, or vesting periods in compensation packages). For example, on July 8 2003 Microsoft announced that employees would receive
common stock associated with a minimum holding period of five years. Kole (1997) documents
that in her sample the minimum holding period before any shares can be sold ranges from 31 to 74
months. In addition, for more than a quarter of the plans the stock cannot be sold before retirement.
5
How does risk aversion affect investment decisions? Within the present paper
the decision maker is risk averse and faces incomplete markets. By investing in the
project, he gives up a risk-free cash flow stream rI and gets in return an undiversifiable
risky cash flow stream X. Thus, his problem is to select the investment time τ that
solves the following problem:
∙Z τ
¸
Z ∞
−ρs
−ρs
u (x) := sup E
e U (rI) ds +
e U (Xs ) ds .
τ
τ
0
This optimization problem can also be written as
U (rI)
+ F (x) .
ρ
u (x) :=
where
F (x) = sup E
τ
∙Z
∞
−ρs
e
τ
¸
(U (Xs ) − U(rI)) ds .
This specification shows that the indirect utility of the decision maker is the sum of
the utility he would derive ignoring the investment option plus the expected change
in utility due to the exercise of the real option.
Denote by β and γ the positive and negative roots of the quadratic equation
1 2
σ ξ (ξ − 1) + µξ − ρ = 0.
2
(5)
Using the strong Markov property of Brownian motion, we can write
£
¤
F (x) = sup E e−ρτ V (Xτ )
τ
where
2
V (x) = 2
σ (β − γ)
½ Z
xγ
0
x
−γ−1 b
s
β
U (s) ds + x
Z
x
∞
¾
U (s) ds ,
−β−1 b
s
(6)
b (s) , U (s)−U(rI). As in the risk-neutral case, X is a sufficient statistic for the
for U
investment surplus and this surplus is increasing in X. Thus the value-maximizing
investment policy can be described by a first passage time τ of the state variable
(Xt )t≥0 to a constant threshold X ∗ . Solving for the utility maximizing threshold
yields the following result.
6
Theorem 1 Consider a risk averse decision maker with an increasing, concave and
once continuously differentiable utility function. Suppose that he can give up a riskfree cash flow stream rI and get in return an undiversifiable risky cash flow stream
X with dynamics governed by (1). Then the indirect utility of the manager satisfies
³ x ´β
1
u (x) := U (rI) + V (X ∗ )
, x < X ∗,
ρ
X∗
where V (.) is defined in (6) and the utility maximizing investment rule is to invest as
soon as X reaches the threshold X ∗ defined by
Z X∗
s−γ−1 [U (s) − U(rI)] ds = 0.
0
Theorem 1 provides the utility maximizing investment rule for any increasing,
concave and once continuously differentiable utility function. To derive specific implications regarding the impact of risk aversion on investment decisions, the model
has to be specified further. Below we examine these implications by considering a special class of utility functions whose relative risk aversion in consumption is a positive
constant:
( 1−R
c
−1
;
R > 0, R 6= 1,
1−R
U (c) =
log (c) ;
R = 1.
In this specification, the constant R is the manager’s relative risk aversion. A simple
application of the result in Theorem 1 yields the following Proposition.
Proposition 2 Assume that the decision maker has power utility with constant relative risk aversion R and that the constant ∆ = ρ + (R − 1) (µ − 0.5Rσ2 ) is strictly
positive. Then, the indirect utility of the manager satisfies
∙
¸³ ´
1
1
1
x β
∗
u (x) := U (rI) +
, x < X ∗,
U (X ) − U (rI)
∗
ρ
∆
ρ
X
where the utility maximizing investment rule satisfies
¶ 1
µ
∆ 1−R
β
∗
X (R) =
r I;
R > 0, R 6= 1.
β−1+R ρ
For the log investor, this utility maximizing investment threshold is given by
µ
¶
1 µ − σ 2 /2
∗
X = r I exp
−
;
R = 1,
β
ρ
which is the limit as R tends to 1 of X ∗ (R).
7
Proposition 2 shows that with power utility the investment threshold selected by
the manager has the same functional form as the one that maximizes project value.
Specifically, the minimum asset value triggering investment is equal to the product of
the cost of investment I and a factor that represents the value of waiting to invest.
While the expression for the value-maximizing investment threshold is familiar from
the real options literature, it is important to note that, within the present model,
this expression reflects the attitude of the manager towards risk. In the particular
case where the manager is risk neutral, we have R = 0 and ρ = r and the investment
threshold is given by
X ∗ (0)
β
=
I
r−µ
β−1
which is the solution reported in equation (4) above.
Proposition 2 also shows that the indirect utility of the manager is equal to the subjective value of perpetual stream of consumption rI plus the change in the subjective
value of this stream associated with the investment decision. Using the expressions
reported in Proposition 2, we get
¸
∙
1 − R ³ x ´β
1
u (x) := U (rI) 1 +
ρ
β − 1 + R X∗
where the second term in the square brackets measures the increase in indirect utility
due to the exercise of the option. (Note that when R > 1, we have U (rI) < 0 and
the subjective option value is still positive.)
4
Model implications
The solution to the model presented in Proposition 2 yields a number of novel implications regarding invetsment policy. These implications are grouped in three categories
as follows.
Risk aversion and the option value to wait. One of the major contributions of
the real options literature is to show that with uncertainty and irreversibility, there
exists a value of waiting to invest. Thus, the decision maker should only invest when
the asset value exceeds the investment cost by a potentially large option premium
8
[see e.g. Dixit and Pindyck (1994)]. As shown in Proposition 1, this incentive to
delay investment is magnified by risk aversion. To better understand this incentive
to delay investment, one has to recall that by investing the entrepreneur transforms
a safe asset into a risky one. As a result, investment exposes him to undiversifiable
cash flow risk. The associated increase in the volatility of consumption leads to a
reduction in the manager’s indirect utility, which in turn provides the entrepreneur
with an incentive to delay investment. This effect is illustrated by Figure 1 which
plots the investment threshold as a function of the relative risk aversion coefficient R
and the volatility of the cash flow process σ for ρ = 0.2 and µ = 0.1.
[Insert Figure 1 Here]
Risk aversion and project value. The above analysis shows that a risk-averse
manager has an incentive to delay investment in comparison with the value-maximizing
policy for well diversified shareholders. As a result, risk aversion induces a reduction
in project value which is equal to the difference between current firm (option) values
under project- and utility-maximizing policies. Denote by Π (Xt ) the present value of
a cash flow steam Xt starting at time t. Assuming that agents are risk neutral, this
present value is given by
Π (Xt ) =
Xt
;
r−µ
for µ < r.
As a proportion of the value of the project under the value-maximizing policy, the
reduction ∆F in project value is then given by:
Π (X ∗ (R)) − I
F (x; X ∗ (R))
=
1
−
∆F = 1 −
F (x; X ∗ (0))
Π (X ∗ (0)) − I
µ
X ∗ (0)
X ∗ (R)
¶β
,
where X ∗ (R) is the investment threshold selected by the decision maker and X ∗ (0)
is the value-maximizing investment threshold.
When R = 3 and σ = 0.3, risk aversion reduces the value of the project by 15%.
Thus, risk aversion delays investment and has a significant impact on the value of
investment projects. To get more insights on the impact of the various parameters of
the model, Figure 2 plots the reduction in project value as a function of the relative
9
risk aversion coefficient R and the volatility of the cash flow process σ for ρ = 0.2
and µ = 0.1.
[Insert Figure 2 Here]
Consistent with economic intuition, Figure 2 shows that the reduction in project value
should increase with both volatility and the coefficient of risk aversion. Interestingly,
this reduction in project value results from two opposite effects. On the one hand, risk
aversion increases the investment threshold and hence the surplus from investment
at the time of investment. On the other hand, risk aversion delays investment and
reduces the probability of investment. This is to this second effect that we now turn.
Probability of investment. The impact of risk-aversion on investment policy can
be analyzed by examining the change in the probability of investment due to risk
aversion. Define the running maximum of the process (Xt )t≥0 at time t by Xsup (t) =
sup0≤τ ≤t X (τ ). The probability of investment over the time interval [0, T ] satisfies:
¸ ³ ´ 2µ ∙
¸
y σ2
− ln (y/X0 ) + µT
− ln (y/X0 ) − µT
√
√
+
,
N
P (Xsup (T ) ≥ y) = N
X
σ T
σ T
∙
where N is the Standard Normal cumulative distribution function, µ = µ − σ 2 /2,
y = X ∗ (R) under the utility-maximizing investment policy, and y = X ∗ (0) under
the value-maximizing investment policy. When R = 6, ρ = 0.2 and µ = 0.1, the
probability of investment over a 5 year horizon is 76% under the value-maximizing
investment policy and 50% under the utility-maximizing investment policy. Thus,
risk aversion has a significant impact on the likelihood of investment. This effect is
also illustrated by Figure 3 below, which plots the probability of investment over a
five-year horizon as a function of the risk aversion coefficient of the decision maker
and the volatility of cash flows.
[Insert Figure 3 Here]
As shown by the figure, the more uncertain is the environment of the decision maker,
the bigger is the impact of risk aversion on the probability of investment.
10
5
Conclusion
Since the seminal papers by Brennan and Schwartz (1985) and McDonald and Siegel
(1986), the literature analyzing investment decisions as options on real assets has
developed substantially. In this literature, it is typically assumed that agents are
risk neutral or that markets are complete and frictionless, so that decisions are made
in a preference-free environment. Yet, in most situations, managers face incomplete
markets either because the cash flows from the firm’s projects are not spanned by those
of existing assets or because of compensations packages that restrict their portfolios.
The generalization of the real option approach to include risk aversion provides
very different implications from standard settings in which agents are risk neutral or
capital markets are frictionless. Notably, we demonstrate that risk aversion provides
an incentive for decision makers to delay investment. As shown in the paper, this
incentive to invest late significantly reduces the probability of investment over a given
horizon and erodes the value of investment projects.
11
References
Brennan, M., and E. Schwartz, 1985, “Evaluating Natural Resource Investments”,
Journal of Business 58, 137-157.
Dixit, A. and R. Pindyck, 1994, Investment Under Uncertainty, Princeton, NJ:
Princeton University Press.
Henderson, V., 2004, “Valuing the Option to Invest in Incomplete Markets,” Working
Paper, Princeton University.
Hugonnier, J., E. Morellec, and S. Sundaresan, 2005, “Growth Options in General Equilibrium: Some Asset Pricing Implications,” Working Paper, Columbia
University.
Kole, S., 1997, “The Complexity of Compensation Contracts,” Journal of Financial
Economics 43, 79-104.
McDonald, R., and D. Siegel, 1986, “The Value of Waiting to Invest”, Quarterly
Journal of Economics 101, 707-728.
Miao, J. and N. Wang, 2004, “Investment, Hedging, and Consumption Smoothing”,
Working Paper, Boston University.
Moel, A., and P. Tufano, 2002, “When Are Real Options Exercised? An Empirical
Investigation of Mine Closings”, Review of Financial Studies 15, 35-64.
Parrino R., A. Poteshman, and M. Weisbach, 2002, “Measuring Investment Distortions when Risk-Averse Managers Decide Whether to Undertake Projects”,
Working Paper, University of Texas, Austin.
Pindyck, R., 1988, “Irreversible Investment, Capacity Choice, and the Value of the
Firm”, American Economic Review 78, 969-985.
12
Figure 1: Selected investment threshold. Figure 1 plots the investment threshold
as a function of the relative risk aversion coefficient R and the volatility of cash flows σ .
Investment threshold
Investment threshold
1.5
1.45
1.4
1.35
1.3
1.25
1.2
0
1
2
Relative
3
4
risk aversion
R
5
1.8
1.7
1.6
1.5
1.4
1.3
1.2
6
0.16
0.18
0.2 0.22 0.24 0.26
Volatility
σ
0.28
0.3
Change in Project Value
Change in Project Value
Figure 2: Change in project value. Figure 2 plots the change in project value as
a function of the relative risk aversion coefficient R and the volatility of cash flows σ .
0.14
0.12
0.1
0.08
0.06
0.04
0.02
0
0
1
2
Relative
3
4
risk aversion
R
5
0.14
0.12
0.1
0.08
0.06
0.04
0.02
0
0.16 0.18
6
0.2 0.22 0.24 0.26
Volatility
σ
0.28
0.3
Probability of Investment
Probability of Investment
Figure 3: Probability of investment. Figure 3 plots the probability of investment
as a function of the relative risk aversion coefficient R and the volatility of cash flows σ .
0.75
0.7
0.65
0.6
0.55
0.5
0
1
2
Relative
3
4
risk aversion
5
6
0.8
0.7
0.6
0.5
0.4
0.16 0.18
R
13
0.2 0.22 0.24 0.26
Volatility
σ
0.28
0.3